Statistical Inference for High-Dimensional Robust Linear Regression Models via Recursive Online-Score Estimation

Abstract: This paper introduces a novel framework for estimation and inference in penalized M-estimators applied to robust high-dimensional linear regression models. Traditional methods for high-dimensional statistical inference, which predominantly rely on convex likelihood-based approaches, struggle to address the nonconvexity inherent in penalized M-estimation with nonconvex objective functions. Our proposed method extends the recursive online score estimation (ROSE) framework of Shi et al. (2021) to robust high-dimensional settings by developing a recursive score equation based on penalized M-estimation, explicitly addressing nonconvexity. We establish the statistical consistency and asymptotic normality of the resulting estimator, providing a rigorous foundation for valid inference in robust high-dimensional regression. The effectiveness of our method is demonstrated through simulation studies and a real-world application, showcasing its superior performance compared to existing approaches.